Note that for \(U_{i_k} = I_{n}\) , RCPG recovers algorithm (4.8) in [1]}, while for \(f \equiv 0\)  we get an algorithm similar to the one in [2]}. However, [1]} derives rates only in the convex setting, while [2]} presents only assymptotic convergence results. In RCPG algorithm we can choose a larger stepsize when \(\psi \)  is convex along coordinates (see Definition ), since \(H_{f,U_{i_k}}\)  must satisfy in this case \(H_{f,U_{i_k}}> L_{i_{k}}/2\) . Otherwise, we should choose \(H_{f,U_{i_k}}> L_{i_{k}}\) .
Consider \(\phi _{i_{k}}^{x_{k}}\)  defined in (REF ).
Then, our algorithm requires computation of the proximal operator of the function \( \phi _{i_{k}}^{x_{k}}\)  at \(U^{T}_{i_k} \nabla f(x_{k})\) :
